= Solution
For a twice differentiable function, <convexity> on $\mathbb R^2$ is equivalent to its <Hessian matrix> being <positive semidefinite> everywhere. Here the <Hessian matrix> is constant:
$$
H=\begin{pmatrix}4&\alpha\\\alpha&2\end{pmatrix}.
$$
Its diagonal entries are positive, so <positive semidefiniteness> is equivalent to $\det H=8-\alpha^2\geq0$. \b[The required parameter range is]
$$
\boxed{-2\sqrt2\leq\alpha\leq2\sqrt2.}
$$
The endpoints give <convex functions> with one flat direction; strict inequalities give a <Hessian matrix> defining a <positive-definite quadratic form> and a <strictly convex function>.
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